Worksheets for GCSE Mathematics. Sequences & Patterns. Mr Black's Maths Resources for Teachers Grades 1-9. Algebra

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1 Worksheets for GCSE Mathematics Sequences & Patterns Mr Black's Maths Resources for Teachers Grades 1-9 Algebra

2 Sequences and Patterns Worksheets Contents Differentiated Independent Learning Worksheets Continuing and Describing Sequences Coordinates in Four Quadrants Generating a Sequence from the Nth Term Deriving the Nth Term of a Linear Sequence Nth Term of Quadratic Sequences Geometric Sequences Recurrence Formulae Page 20 Page 30 Page 40 Page 50 Page 60 Page 70 Page 80 Solutions Continuing and Describing Sequences Page 9 Coordinates in Four Quadrants Page 10 Generating a Sequence from the Nth Term Page 11 Deriving the Nth Term of a Linear Sequence Page 12 Nth Term of Quadratic Sequences Page 13 Geometric Sequences Page 14 Recurrence Formulae Page 15

3 Q1. Match the sequence with its description. Continuing and Describing Sequences 5, 8, 11, 14, 17, 20 5, 10, 20, 40, 80, 160 Start at 5 and add 3 Start at 5 then subtract one 5, 4, 3, 2, 1, 0 Start at 5 double each term 5, 10, 15, 20, 25, 30 First six multiples of five Q2. Describe in words and continue each of these sequences. a) b) c) d) e) f) g) h) i) Q3. Fill in the missing terms in each sequence. a) b) c) d) e) f) g) h) i) Q4. Draw the next two pictures in each sequence and describe the patterns you notice. a) b)

4 Q1. Write down the coordinates of the following points: Coordinates in Four Quadrants A = F = B = G = C = H = D = I = E = Q2. a) ABEQ is a rectangle. Plot the coordinate Q.. b) GFDR is a square. Plot the coordinate R c) S is the midpoint of GF. What are the coordinates of S? d) T is the midpoint of CI. What are the coordinates of T? e) Which two coordinates make the line GH into two different right-angled triangles? f) Point U centre of the rectangle ABEQ. State its coordinates. Q3. a) Join (0,0), (3,0), (0,2), (0,0) Shape = b) Join (0,0), (-2,0), (-5,2), (-5,0), (0,0) Shape = c) Join ( -4,-3), (2,-3), (3,-1), (-3,-1), (-4,-3) Shape =

5 Generating a Sequence from the Nth Term Q1. Generate the first six terms of each sequence. a) SSSSSS = nn + 3 b) SSSSSS = 5nn c) SSSSSS = 2nn + 3 d) SSSSSS = nn 3 e) SSSSSS = 12 nn f) SSSSSS = 5nn + 8 g) SSSSSS = 4nn 1 h) SSSSSS = 7nn 2 i) SSSSSS = 2(nn + 6) j) SSSSSS = 7(8 nn) k) SSSSSS = (nn + 9) l) SSSSSS = 5(11 3nn) Q2. Generate the first six terms of each sequence. a) SSSSSS = nn b) SSSSSS = nn 2 4 c) SSSSSS = nn 2 + nn d) SSSSSS = nn 2 2nn e) SSSSSS = nn 2 + nn + 4 f) SSSSSS = nn 2 + 5nn g) SSSSSS = 2nn 2 h) SSSSSS = nn 3 i) SSSSSS = nn 3 7 j) SSSSSS = nn k) SSSSSS = nn 3 2nn l) SSSSSS = 6nn 3 Q3. Match the term to the sequence. 36, 24, 12, 0, , 42, 23, -14, -75 SSSSSS = SSSSSS = , 13, 16, 19, 21 17, 14, 11, 8, 5 SSSSSS = nn SSSSSS = 44( ) 7, 10, 15, 22, 31 SSSSSS = 5555 nn 33 Q4. Generate the first four terms of each sequence. a) SSSSSS = 1 4nn+7 d) SSSSSS = nn2 + 4nn nn + 4 b) SSSSSS = 2nn+1 3nn+5 e) SSSSSS = 2 nn g) SSSSSS = (3 nn ) h) SSSSSS = 2 nn

6 Q1. Derive the nth term for each of these sequences: Deriving the Nth Term of a Linear Sequence a) n Sequence b) n Sequence c) n Sequence d ) n Sequence Q2. Derive the nth term for each of these sequences: a) b) c) d) e) f) g) h) i) j) k) l) Q3. Derive the nth term for each of these sequences: a) b) c) d)

7 Nth Term of Quadratic Sequences Q1. Derive the nth term for each of these sequences: a) n Sequence b) n Sequence c) n Sequence d ) n Sequence Q2. Derive the nth term for each of these sequences: a) b) c) d) e) f) g) h) i) j) k) l) Q3. Derive the equation of each parabola. a) b) c)

8 Geometric Sequences Q1. Which of the following sequences are geometric? i) For the ones that are, state the value of the common ratio. ii) Continue the geometric sequences for two more terms. a) 6, 12, 24, 48,96, 192, b) 0.25, 0.5, 0.75, 1, 1.25, 1.5, c) 5, 15, 25, 35, 45, 55, d) a, 2a, 3a, 4a, 5a, 6a, e) 2, -1, 0.5, -0.25, 0.125, , f) c, c 2, c 3, c 4, c 5,. Q2. Match the geometric progression with the nth term. 5, 10, 20, 40, 80, n 1 1, 5, 25, 125, 625, n 1 2, 10, 50, 250, 1250, n 1 1, 2, 4, 8, 16, 32, n 1 Q3. Find i) the nth term and ii) the 10 th term of each geometric progression. a) 6, 18, 54, 162, 486, 1458, b) 10, 20, 40, 80, 160, 320, c) 2, 8, 32, 128, 512, 2048, d) 6, 18, 54, 162, 486, 1458, e) 81, 27, 9, 3, 1, 1 3, f) 1, 1 2, 1 4, 1 8, 1 16, 1 32, g) 64, 16, 4, 1, 1 4, 1 16 h) 8, -16, 32, -64, 128, -256 Q4. a) The 3 rd term of a geometric sequence is 15 and the 4 th term is 75. i) What is the common ratio? ii) What is the first term of the geometric sequence? b) The first term in a geometric sequence is 4 and the third term is 1. The common ratio is greater than 0. i) What is the common ratio? ii) What is the fifth term of the geometric sequence? c) The 2 nd term of a geometric sequence is 104 and the 5 th term is 13. i) What is the common ratio? ii) What is the first term of the geometric sequence?

9 Recurrence Sequences Q1. Match the recurrence formula with its generated sequence. u n+1 = 2 + 3U n, U 1 = 2 2, 5, 8, 11, 14, 17 u n+1 = 3(U n + 2), U 1 = 2 2, 6, 18, 54, 162, 486 u n+1 = 3U n, U 1 = 2 2, 7, 17, 37, 77, 157 u n+1 = 3 + U n, U 1 = 2 2, 8, 26, 80, 242, 728 u n+1 = 2U n + 3, U 1 = 2 2, 12, 42, 132, 402 Q2. Find the first five terms of the following recurrence relationships a) u n+1 = U n + 5, U 1 = 2 b) u n+1 = 3U n 4 U 1 = 4 c) u n+1 = 2(1 + U n ), U 1 = 1 d) u n+1 = U n 3, U 1 = 6 e) u n+1 = U n 2 5, U 1 = 5 f) u n+1 = U n U n + 2, U 1 = 8 g) u n+1 = U n, U 1 = 64 h) i) u n+1 = 1 2U n, U 1 = 5 u n+1 = 1 U n 2U n + 3, U 1 = 2 5 Q3. Suggest possible recurrence relationships for the following sequences. a) 2, 6, 10, 14, 18, 22, b) 15, 12, 9, 6, 3, 0, -3, c) 3, -1, 3, -1, 3, -1, 3, d) 0.5, 1, 2, 4, 8, 16, 32,... e) 512, 256, 128, 64, 32, 16, f) 1, 1, 1, 1, 1, 1, 1, 1, 1, Q4. A recurrence sequence is defined as U n+2 = au n+1 + 2U n where a is a constant. a) Write an expression for U 3 when U 1 = 2 and U 2 = 5. b) Write an expression for U 4 c) U 3 has the value 41 Find the value of U 4

10 Solutions Continuing and Describing Sequences Q1. 5, 8, 11, 14, 17, 20 5, 10, 20, 40, 80, 160 Start at 5 and add 3 Start at 5 then subtract one 5, 4, 3, 2, 1, 0 Start at 5 double each term 5, 10, 15, 20, 25, 30 First six multiples of five Q2. a) Start a one, add four b) Start at twelve, subtract two c) Start at eight, add seven d) Start at nine, subtract one e) Start at one, add ten f) Start at twelve, add three g) Start at eleven, subtract five h) Start at five, add eight i) Start at one, subtract two Q3. a) b) c) d) e) f) g) h) i) Q4. a) Adding a square on the top and right sides. b) Adding a square on each protruding edge.

11 Solutions Generating a Sequence from the Nth Term Q1. a) 4, 5, 6, 7, 8, 9 b) 5, 10, 15, 20, 25, 30 c) 5, 7, 9, 11, 13, 15 d) -2, -3, -4, -5, -6, -7 e) 11, 10, 9, 8, 7, 6 f) 13, 18, 23, 28, 33, 38 g) 3, 7, 11, 15, 19, 23 h) 5, 12, 19, 26, 33, 40 i) 14, 16, 18, 20, 22, 24 j) 49, 42, 35, 28, 21, 14 k) -10, -11, -12, -13, -14, -15 l) -40, -25, -10, 5, 20, 35 Q2. a) 4, 7, 12, 19, 28, 39 b) -3, 0, 5, 12, 21, 32 c) 2, 6, 12, 20, 30, 42 d) -1, 0, 3, 8, 15, 24 e) 6, 10, 16, 24, 34, 46 f) 6, 14, 24, 36, 50, 66 g) 2, 8, 18, 32, 50, 72 h) 1, 8, 27, 64, 125, 216 i) -6, 1, 20, 57, 118, 209 j) 5, 12, 31, 68, 129, 220 k) -1, 4, 21, 56, 115, 204 l)6, 48, 162, 384, 750, 1296 Q3. 36, 24, 12, 0, , 42, 23, -14, -75 SSSSSS = SSSSSS = , 13, 16, 19, 21 17, 14, 11, 8, 5 SSSSSS = nn SSSSSS = 44( ) 7, 10, 15, 22, 31 SSSSSS = 5555 nn 33 Q4. a) b) d) 1, 2, 3, 4 e) 2, 4, 8, 16 g) -3, -9, -21,-81 h) 0.5, 0.25, 0.125,

12 Solutions Deriving the Nth Term of a Linear Sequence Q1. a) Sequence = 5n + 4 b) Sequence = 5n + 10 c) Sequence = 2n + 5 d) Sequence = 22 2n Q2. a) 5n - 6 b) 12n + 2 c) 12n - 8 d) 11n + 6 e) 8n + 6 f) n - 1 g) 15 2n h) 8n + 5 i) 11n + 1 j) 0.5n - 2 k) 1.5n - 2 l) -2.5n - 2 Q3. a) nn+6 4nn+4 c) 7nn+6 nn+4 b) 2nn 4nn+1 d) 8 2nn 4nn+5

13 Q1. Coordinates in Four Quadrants A = (-3,5) F = (4,0) B = (3,4) G = (0,0) C = (-3,-2) H = (0,-4) D = (3,-3) I = (-5,-4) E = (-3,1) Q2. a) Coordinate Q = (1,2) b) Coordinate R = (0,-4) c) Coordinate S = (4,-2) d) Coordinate T = (-3,-1) e) GHF & GHI f) Coordinate U = (-1.5, 3.5) Q3. a) Shape = Right-Angled Triangle b) Shape = Trapezium c) Shape = Parallelogram

14 Nth Term of Quadratic Sequences Solutions Q1. a) Sequence = n b) Sequence = n 2-2 c) Sequence = 2n 2 d) Sequence = 2n Q2. a) n 2 n + 3 b) n 2-3n - 10 c) n 2 + n + 2 d) n 2-3n - 7 e) 2n 2-4n f) 2n 2-5n - 9 g) 3n 2 + 4n - 5 h) 2n 2-2n - 7 i) 3n 2-2n - 2 j) 3n 2 + 2n - 10 k) 4n 2 + 3n + 7 Q3. a) yy = xx 2 3 b) yy = xx 2 3 c) yy = xx 2 + 2

15 Solutions Geometric Sequences Q1. a) e) f) i) r = 2, i) r = -0.5 i) r = c ii)384, 768 ii) , ii) c 6, c 7 Q2. 5, 10, 20, 40, 80, n 1 1, 5, 25, 125, 625, n 1 2, 10, 50, 250, 1250, n 1 1, 2, 4, 8, 16, 32, n 1 Q3. a) b) c) d) i) 6 3 n 1 i) 10 2 n 1 i) 2 4 n 1 i) n 1 ii) ii) 5120 ii) ii) e) f) g) h) i) ii) n 1 i) 1 n 1 2 ii) i) n 1 ii) i) 8 2 n 1 ii) Q4. a) i) r = 5 ii) 0.6 b) i) 0.5 ii) 0.25 c) i) 1 2 ii) 208

16 Recurrence Sequences Solutions Q1. u n+1 = 2 + 3U n, U 1 = 2 2, 5, 8, 11, 14, 17 u n+1 = 3(U n + 2), U 1 = 2 2, 6, 18, 54, 162, 486 u n+1 = 3U n, U 1 = 2 2, 7, 17, 37, 77, 157 u n+1 = 3 + U n, U 1 = 2 2, 8, 26, 80, 242, 728 u n+1 = 2U n + 3, U 1 = 2 2, 12, 42, 132, 402 Q2. Find the first five terms of the following recurrence relationships a) 2, 7, 12, 17, 22 b) 4, 8, 20, 56, 164 c) -1, 0, 2, 6, 14 d) 6, 2, 2 3, 2 9, 2 27 e) -5, 20, 395, , f) 8, 4 5, 2 7, 1 8, 1 17 g) 64, 8, 2 2, 1.682, h) 5, 1 10, 5, 1 10, 5, 1 10 i) 2 5, 16 63, , , , Q3. a) u n+1 = U n + 4, U 1 = 2 b) u n+1 = U n 3, U 1 = 15 c) u n+1 = 2 U n, U 1 = 3 d) u n+1 = 2U n, U 1 = 0.5 e) u n+1 = U n 2, U 1 = 512 f) u n+1 = U n 2, U 1 = 1 Q4. a) U 3 = 5a + 6 b) U 4 = 5a 2 + 6a + 10 c) U 4 = 297